Definition and Properties

Differential Geometry of 3D Surfaces

Let us first recall briefly some results of differential geometry about surface curvatures (a good introduction to these notions can be found in [8] or in [20]). In this paper, we call a smooth surface a 3D surface that is continuously differentiable up to the third order. At any point P of such a 3D surface, we can define one curvature per direction t in the tangent plane of the surface. This directional curvature kt is the curvature of the 3D curve defined by the intersection of the plane (P, t, n) with the surface, where n is normal to the surface.

Except for the points where this curvature kt is the same for all the directions t, which are called umbilic points, the total set of curvatures can be described with only two privileged directions, t1 and t2, and two associated curvature values, k1 = kti and k2 = k^, which are called respectively the principal directions and the associated principal curvatures of the surface at point P, as shown in Fig. 1. These two principal curvatures are the extrema of the directional curvatures at point P, and (except for umbilic points) one of these two is maximal in absolute value, let us say k1: We call this the largest curvature, in order not to mistake it for the maximal curvature. We simply call the second (principal) curvature the other principal curvature k2.

Extremal Lines

The crest lines are intuitively the loci of the surface where the "curvature" is locally maximal. More precisely, we define them as the loci of the surface where the largest curvature, k1, is locally maximal (in absolute value) in the associated principal direction t1. In [26], it is shown that these points can be defined as the zero-crossing of an extremality function e, which is the directional derivative of k1 in the direction t1.

We have proposed another method to compute them in [37,38], for the case of isointensity surfaces. Our method is based on the use of the implicit functions theorem. Basically, we have shown that the crest lines can be extracted as the intersection of two implicit surfaces f = I and e = 0, where f represents the intensity value of the image, I an isointensity threshold, and e = Vk1 • 11 is the extremality function (see Fig. 2, left). We have proposed an algorithm, called the Marching Lines, to automatically extract these crest lines. This algorithm can also be used to overcome some orientation problems (mainly due to the fact that the principal directions are only directions and not oriented vectors), by locally orienting the principal directions along the extracted lines.

In fact, for each point of the surface, two different extremality coefficients can be defined, corresponding to the two principal curvatures:

We found experimentally that the maxima (in absolute values) are more stable landmarks than the minima: Crests or rifts (maxima) are stable, whereas the loci in a valley where the ground floor is the flattest (minima) are very sensitive to small perturbations in the data.

We call extremal lines all the lines defined as the zero-crossings of either e1 or e2. There are therefore four major different types of extremal lines, depending on whether the corresponding curvature is the largest or the second one and whether it is a local maximum or minimum. Furthermore, the

P ane o

AlffrculaiiTtg plane

Curve invariant*: k. t fkiii-c maximal curvature

Snriyti; invariants: L, L

FIGURE 1 Differential geometry of 3D curves and surfaces. (Left) Principal directions and curvatures of a surface. (Right) Frenet trihedron of a 3D curve and first differential-invariants: curvature and torsion.

AlffrculaiiTtg plane

P ane o

Curve invariant*: k. t fkiii-c maximal curvature

Snriyti; invariants: L, L

FIGURE 1 Differential geometry of 3D curves and surfaces. (Left) Principal directions and curvatures of a surface. (Right) Frenet trihedron of a 3D curve and first differential-invariants: curvature and torsion.

FIGURE 2 (Left) Crest lines as the intersection surfaces. (Right) Definition of the extremal points as the intersection of three implicit surfaces. Adapted from J.-P. Therion. New feature points based on geometric invariants for 3D image registration. Lnt. J. Comp. Vis., 18(2):121-137, 1996, with permission.

FIGURE 2 (Left) Crest lines as the intersection surfaces. (Right) Definition of the extremal points as the intersection of three implicit surfaces. Adapted from J.-P. Therion. New feature points based on geometric invariants for 3D image registration. Lnt. J. Comp. Vis., 18(2):121-137, 1996, with permission.

signs of the largest and second curvatures help to distinguish between four additional subtypes of extremal lines, leading to a classification into 16 types. The crest lines are two of them: positive largest curvature maxima (k1 >0 and Ve: • t1 <0) and negative largest curvature minima (k1 <0 and Ve1 • t1 >0).

Extremal Points

We now define the extremal points as the intersection of the three implicit surfaces: f = I, e1 = 0, and e2 = 0. The notions of extremal lines and extremal points are closely related to the notion of corner points, in 2D images, as defined in [21], [27], and [9]. A study of the evolution in two dimensions of corner points with respect to the scale can be found in [14]. A similar study on the scale-space behavior of the extremal lines and the extremal points was presented in [12].

Extremalities e1 and e2 are geometric invariants of the implicit surface f = I: They are preserved with rigid transforms (rotations and translations of the object). Therefore, the relative positions of the extremal points are also invariant with respect to a rigid transformation, i.e, for two different acquisitions of the same subject. There are 16 different types of extremal points, depending on the type of extremality: local minimum or maximum of the extremalities e1 and e2 and the signs of k1 and k2. This classification can be used to reduce the complexity of the matching algorithm.

However, the intuitive interpretation of extremal points is not straightforward. The extremal lines are 3D curves, for which we are able to compute the curvature, but the extremal points are generally not the points of the extremal lines whose curvature is locally maximal. Even if they are not extremal curvature points, the extremal points are very well defined, and there is no reason for their locations along the extremal lines to be less precise that the lines positions themselves, because the precision of the computation of k1 and k2 is approximately the same.

Geometric Characteristics

Let us begin with the points on a surface. We have already seen (Fig. 1, left) that any such point could be provided with a trihedron (n, t1; t2) formed by the normal to the surface and the two principal directions. As our points are also on extremal lines, we could provide them with the differential characteristics of 3D curves (Fig. 1, right), i.e., the Frenet trihedron (t, nc, b), where t is the tangent to the curve, nc its normal, and b the binormal.

These two trihedrons are not the same, as the extremal lines are generally not lines of curvature. However, as the curve is embedded in the surface, the tangent to the curve t is constrained to be in the tangent plane of the surface spanned by (t 1; t2). Thus, there are two independent parameters characterizing the relative configuration of the trihedron: we can measure two angles 6 = (t, t1) and ^ = (n^fn). These characteristics are invariant with respect to rigid transformations.

Two other invariants come from the surface ( principal curvatures k1 and k2). One could also think to add the curvature k, the torsion t, of the curve and the geodesic torsion xg of the curve with respect to the surface, but it appears that k and Tg are completely determined by the surface invariants: k cos ^ = k1 cos2 6 + k2 sin2 6 and Tg = (k2 — k1) cos 6 sin 6. Thus, we are left with the torsion of the curve.

However, the computation of the Frenet trihedron (t, g, b) and the curve torsion t has to be done on the extremal curve itself after its extraction. If this can be done directly on the polygonal approximation, a much better method is to compute the characteristics on a local B-spline approximation of the curve [13].

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